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O (n) set of all orthogonal U T =U 1 (therefore real) matrices of dimension n×n Rotations in 3-dim space SO(3) 4-dim space-time Lorentz transformations SO(4) Orbital angular momentum rotations SO( ℓ ) (mixing of quantum mechanical states) cos sin 0 Rz( ) = -sin cos U (n) set of all n×n UNITARY matrices U † =U 1 i.e. U † U SU(n) “special” unimodular subset of the above det(U)=1 SO (n) “special” subset of the above: unimodular, i.e., det(U)=1 group of all rotations in a space of n -dimensions ALL known “external” space-time symmetries in physics new “internal” symmetries (beyond space-time)
Citation preview
/
/1 Gi
Gji
eeHn
jj
We’ve found an exponential expression for operators
n number of dimensions ofthe continuous parameter Generator G
The order (dimensions) of G is the same as H
We classify types of transformations (matrix operator groups) as
Orthogonal O(2) SO(2)O(3) SO(3)
Unitary U(2) SU(2)U(3) SU(3)
groups in the algebraic sense: closed within a defined mathematical operation
that observes the associative propertywith every element of the group having an inverse
O(n) set of all orthogonal UT=U1 (therefore real) matrices of dimension n×n
Rotations in 3-dim space SO(3)4-dim space-time Lorentz transformations SO(4)
Orbital angular momentum rotations SO(ℓ ) (mixing of quantum mechanical states)
cos sin 0Rz() = -sin cos 0
0 0 1
U(n) set of all n×n UNITARY matrices U†=U1
i.e. U†USU(n) “special” unimodular subset of the above det(U)=1
SO(n) “special” subset of the above: unimodular, i.e., det(U)=1group of all rotations in a space of n-dimensions
ALL known“external”space-timesymmetriesin physics
new“internal”
symmetries(beyond
space-time)
SO(3) cos3 sin3 0
-sin3 cos3 0
0 0 1
cos2 0 -sin2
0 1 0 sin2 0 cos2
1 0 0
0 cos1 sin1
0 -sin1 cos1 R(1,2,3)=
cos3cos2+sin3sin2 cos1sin3-sin1sin2sin3 sin1sin3-cos1sin2cos3
-cos2sin3 cos1cos3-sin1sin2sin3 sin1cos3-cos1sin2sin3
sin2 -sin1cos2 cos1cos2
=
Contains SO(2) subsets like:
cos sin 0
-sin cos 0
0 0 1Rz( ) =
acting on vectors like
v =vx
vy
vzin the i, j, k basis^ ^ ^
do notcommute
do commute
NOTICE:all real
and orthogonal
Call this SO(2)
cos sin -sin cos
R v = vx
vy
Obviously “reduces” to a 2-dim representation
What if we TRIED to diagonalize it further?
seek a similarity transformation on the basis set: U†xwhich transforms all vectors: Uv
and all operators: URU†
cos - sin 0 -sin cos - 0 = 00 0 1-
An Eigenvalue Problem
24cos4cos2 2
1coscos 2
2cos1 cos i sin cos i
Eigenvalues: =1, cos + isin , cos isin
(1-[1 - 2cos + ]=0
=1
= (1-[cos2-2cos+sin2]=0
cos sin 0 a a-sin cos 0 b = b 0 0 1 c c
To find the eigenvectors
21
21
2i
2i
a/b = b/a ?? a=b=0
acos + b sin = aasin + b cos = b
c = c
a(cos ) = bsin b(cos ) = asin
for =cos+isin
for =cosisin
for =1
b = i a, c = 0since a*a + b*b = 1
a = b =
b = i a, c = 0since a*a + b*b = 1
a = b =
acos + b sin = a(cos+isin)asin + b cos = b(cos+isin)
c = c(cos+isin)
cos sin 0 0
-sin cos 0 0
0 0 1 0 1 0
With < v | R | v >
eigenvectorsURU†
cos +isin 0 0
= 0 1 0
0 0 sin icos
21
21
2i
2i
cos +isin 0 0
= 0 1 0
0 0 sin icos
and under a transformation to this basis(where the rotation operator is diagonalized)
vectors change to:
v1 (v1+iv2)/Uv = U v2 = v3
v3 (v1iv2)/
2
2
< v | R | v >
SO(3)R(1,2,3)
cos3cos2+sin3sin2 cos1sin3-sin1sin2sin3 sin1sin3-cos1sin2cos3
-cos2sin3 cos1cos3-sin1sin2sin3 sin1cos3-cos1sin2sin3
sin2 -sin1cos2 cos1cos2
=
Contains SO(2) subsets like:
cos sin 0
-sin cos 0
0 0 1Rz( ) =
acting on vectors like
v =vx
vy
vzin the i, j, k basis^ ^ ^
which we just saw can be DIAGONALIZED:
Rv = e+i 0 0 0 1 0 0 0 iei
)(2
1yx ivv
)(2
1yx ivv
zv
Block diagonal form means NO MIXING of components!
Rv = e+i 0 0 0 1 0 0 0 iei
)(2
1yx ivv
)(2
1yx ivv
zv
Reduces to new “1-dim” representation of the operatoracting on a new “1-dim” basis:
e+i )(2
1yx ivv
)cossin(
)sincos(
))(sin(cos
21
21
yx
yx
yx
vvi
vv
ivvi
iei )(2
1yx ivv
)cossin(
)sincos(2
1
yx
yx
vvi
vv
+
i
)( 2121 iii eee321321 )()( iiiiii eeeeee
1 ii ee
R(1) R(2)= R(1+2)
UNITARY now!(not orthogonal…)
ei is the entire set of all 1-dim UNITARY matrices, U(1)
obeying exactly the same algebra as SO(2)
SO(2) is ISOMORPHIC to U(1)
SO(2) is supposed to be the group of all ORTHOGONAL 22 matriceswith det(U) = 1
a bc d
a cb d = a2+b2 ab+bd
ac+bd c2+d2
ac = bd a2 + b2 = c2 + d2 = and
along with: det(U) = ad – bc = 1 abd – b2c = b a2c – b2c = b c(a2 + b2) = b c = b
which means:ac = (c)d a = d
a b-b a
SO(2)So all matrices have the SAME form:
a2 + b2 = 1with
i.e., the set of all rotations in the space of 2-dimensionsis the complete SO(2) group!
det(A) n1 n2 n3···nN An11 An22 An33 … AnNN n1,n2,n3…nN
N
some properties
det(AB) = (detA)(detB) = (detB)(detA) = det(BA)
since these are just numbers
which meansdet(UAU†) = det(AU†U) = det(A)
So if A is HERMITIAN it can be diagonalized by a similarity transformation (and if diagonal)
det(A) …(n1 n2 n3···nN An11)An22 An33 … AnNN nN
N
n3
N
n2
N
n1
N
Only A11 term0 only diagonal termssurvive, here that’s A22
det(A) 123…N A11 A22 A33 … ANN
= N
Determinant values do not changeunder similarity transformations!
completely antisymmetric tensor (generalized Kroenicker )
(Akk+Bkk ) k=1
N
another useful property
det(A+B) = (A+B)11 (A+B)22 (A+B)33 … (A+B)NN
= (A11+B11)(A22+B22)(A33+B33)…(ANN+BNN)
If A and B are both diagonal*
(Ak+B
k ) k=1
N
*or are commuting Hermitian matrices
det(A+B+C+D+…) =
so
(Akk+Bkk+Ckk+Dkk+ …) k=1
N
Tr(A) Aii = A11+A22 +A33…+ANNi
N
We define the “trace” of a matrix as the sum of its diagonal terms
Tr(AB) = (AB)ii = AijBji = BjiAiji
N
i
N
j
N
j
N
i
N
= (BA)jj = Tr(BA)
Notice:
Tr(UAU†) = Tr(AU†U) = Tr(A) Which automatically implies: Traces, like
determinantsare invariantunder basis
transformationsSo…IF A is HERMITIAN (which means it can always be diagonalized)
Tr(A) = N
Operators like GieU
if unitary U: GiGi eeU
†† †
) ( GGiGiGi eeeUU
† † † which has to equal
= 1
G = G†
The generators of UNITARY operators are HERMITIANand those kind can
always be diagonalized
Since in general AAAAAAIeU A
!31
!21
In a basis where A is diagonal, so is AA, AAA,… I is already!So U=eA is diagonal (whenever A is)!
What does this digression have to do with the stuff WE’VE been dealing with??
)!3
1!2
1Idet()det(det AAAAAAeU A
N
k
iN
kkkk e
11
32 )!3
1!2
11(
)(Tr3214321 ANN eeeeeee
If U=eiA detU=eiTr(A)
For SU(n)…unitary transformation matrices with det=1
detU = 1 Tr(A)=0
SO(3) is a set of operators (namely rotations) on the basis100
x010
y001
z
zvyvxvv zyx ˆˆˆ
100
010
001
zyx vvv
such that:
z
y
x
z
y
x
vvv
vvv
R preserves LENGTHS and DISTANCES
SU(3) NEW operators (not EXACTLY “rotations”, but DO scramble components) which also act on a 3-dim basis (just not 3-dim space vectors)
p
p +
1951
m=1115.6 MeVmp=938.27 MeV
Look!
By 1953
+ p + m=1115.6 MeV
+ m=1321.0 MeV p +
100
p010
n001
Where a general state (particle) could be expressed
100
010
001
n pnp
N
where NNN GieU
for some set of generators (we have yet to specify)
A model that considered ’s paired composites of these 3 eigenstates
pn pn (n+ n)21
and successfully accounted for the existence, spin, and mass hierachy of +, 0,
K+, K0, K, K0
, ,
ppp ppn pnn nnnunfortunately also predicted the existence of states like:
ppp ppn pnn nnn
none the less NNN GieU
for some set of generators (we have yet to specify)
SU(3)and
means UNITARY The Gi must all be HERMITIAN
and det U = 1 The Gi must all be TRACELESS
As an example consider
SU(2) set of all unitary 2×2 matrices with determinant equal to 1.I claim this set is built with the Pauli matrices as generators!
2/
ieUU which described rotations (in Dirac space) of spinors
0110
x
00i
iy
10
01z
Are these generators HERMITIAN? TRACELESS?
1001
00
0110
i
iDoes
cover ALL possibleHermitian
2×2 matrices?
In other words: Are they linearly independent? Do they span the entire space?
What’s the most general traceless HERMITIAN 22 matrices?
c aiba+ib c
aibaib
cc
and check out:
= a +b +c 0 11 0
0 -ii 0
1 00 -1
They do completely span the space!Are they orthogonal (independent)?
You can’t make one out of any combination of the others!